--- Adjugate Matrix properties ---
I have a small question regarding the adjugate matrix.

Suppose we have a cube (n×n ) singular matrix with a rank of n−1 .

Now I have two questions I'm trying to investigate:

1.Is it possible that the adjugate matrix rank won't be changed? That is, if we have a n×n matrix with a rank of n−1 the adjugate will have the same rank (n−1 ).

2. I know that in case rank of A is n−1 the adj(adj(A)) is 0 . I don't understand why. Is there any relation between these two questions?

Thanks alot, Guy

Question

--- Adjugate Matrix properties ---
I have a small question regarding the adjugate matrix.

Suppose we have a cube (n×n ) singular matrix with a rank of n−1 .

Now I have two questions I'm trying to investigate:

1.Is it possible that the adjugate matrix rank won't be changed? That is, if we have a n×n  matrix with a rank of n−1  the adjugate will have the same rank (n−1 ).

2. I know that in case rank of A is n−1 the adj(adj(A))  is 0 . I don't understand why. Is there any relation between these two questions?

Thanks alot, Guy